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Physical Review A

Solution of the 1,3-contracted Schrödinger equation through positivity conditions on the two-particle reduced density matrix

David A. Mazziotti

Published Dec 12, 2002• 48 citations
Publisher

Abstract

Correlation energies and reduced density matrices (RDMs) of atoms and molecules are directly computed by solving the 1,3-contracted Schr\"odinger equation (1,3-CSE). The solution of the 1,3-CSE synthesizes two optimization strategies recently employed for the direct determination of the 2-RDM: (i) variational minimization of the energy with respect to a 2-RDM constrained by positivity conditions [D. A. Mazziotti, Phys. Rev. A $65,$ 062511 (2002)] and (ii) the contracted power method for solving the 2,4-CSE [D. A. Mazziotti, J. Chem. Phys. $116,$ 1239 (2002)]. While both the 3- and the 4-RDMs in the 2,4-CSE are reconstructed from the 2-RDM by cumulant expansions, similar techniques cannot be directly applied to the 1,3-CSE because constructing the 2-RDM from the 1-RDM with cumulant theory does not improve upon the mean-field approximation. We, however, establish a unique mapping from the 1-RDM to the 2-RDM by searching for the 2-RDM, constrained by contraction and N-representability conditions, which minimizes the energy. The 2-RDM constrained search is practically implemented through recent advances in positive semidefinite programming. With the variational reconstruction of the 2-RDM and a cumulant reconstruction of the 3-RDM, the 1,3-CSE may be solved via a contracted power method for the ground-state energy and RDMs. The initial RDMs, it is shown, need not be N representable for the contracted power method to converge; this allows us to choose the original RDMs from a variational calculation with approximate N-representability conditions on the 2-RDM. Application of the 1,3-CSE algorithm to atoms and molecules yields highly accurate correlation energies both near and far from equilibrium geometries.

DOI
10.1103/physreva.66.062503

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Correlation energies and reduced density matrices (RDMs) of atoms and molecules are directly computed by solving the 1,3-contracted Schr"odinger equation (1,3-CSE). The solution of the 1,3-CSE synthesizes two optimization strategies recently employed for the direct determination of the 2-RDM: (i) variational minimization of the energy with respect to a 2-RDM constrained by positivity conditions [D. A. Mazziotti, Phys. Rev. A 65,65,65, 062511 (2002)] and (ii) the contracted power method for solving the 2,4-CSE [D. A. Mazziotti, J. Chem. Phys. 116,116,116, 1239 (2002)]. While both the 3- and the 4-RDMs in the 2,4-CSE are reconstructed from the 2-RDM by cumulant expansions, similar techniques cannot be directly applied to the 1,3-CSE because constructing the 2-RDM from the 1-RDM with cumulant theory does not improve upon the mean-field approximation. We, however, establish a unique mapping from the 1-RDM to the 2-RDM by searching for the 2-RDM, constrained by contraction and N-representability conditions, which minimizes the energy. The 2-RDM constrained search is practically implemented through recent advances in positive semidefinite programming. With the variational reconstruction of the 2-RDM and a cumulant reconstruction of the 3-RDM, the 1,3-CSE may be solved via a contracted power method for the ground-state energy and RDMs. The initial RDMs, it is shown, need not be N representable for the contracted power method to converge; this allows us to choose the original RDMs from a variational calculation with approximate N-representability conditions on the 2-RDM. Application of the 1,3-CSE algorithm to atoms and molecules yields highly accurate correlation energies both near and far from equilibrium geometries.
The power method for solving N-particle eigenvalue equations is contracted onto the two-particle space to produce a reduced “variational” method for solving the contracted Schrödinger equation (CSE), also known as the density equation. In contrast to the methods which solve a system of approximate nonlinear equations to determine the two-particle reduced density matrix (2-RDM) nonvariationally, the contracted power method updates the 2-RDM iteratively through a “gradient” of the N-particle energy. After each power iteration we modify the 2-RDM to satisfy certain N-representability conditions through an extension of purification to correlated RDMs. The contracted power method is illustrated with a variety of molecules. Significant features of the present calculations include (i) accurate results for both first- and second-order functionals for building the 3- and the 4-RDM’s from the 2-RDM’s; (ii) the first molecular implementation of the Mazziotti correction within the CSE [Mazziotti, Phys. Rev. A 60, 3618 (1999)]; (iii) a spin–orbital formulation; (iv) the treatment of both core and valence orbitals as active; and; (v) a reduction of the CSE computational scaling through fast summation and the natural-orbital transformation.
A contracted Schrödinger equation (1,2-CSE) is derived for the class of Hamiltonians without explicit interactions including those from Hartree–Fock and density functional theories. With cumulant reconstruction of the two-particle reduced density matrix (2-RDM) from the one-particle-RDM (1-RDM), the 1,2-CSE may be expressed solely in terms of the 1-RDM. We prove that a 1-RDM satisfies the 1,2-CSE if and only if it is an eigenstate of the N-particle Schrödinger equation. The 1,2-CSE is solved through the development and implementation of a reduced, linear-scaling analog of the ordinary power method for finding matrix eigenvalues. The power formula for updating the 1-RDM requires fewer matrix operations than the gradient procedure derived by Li et al. [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)]. Convergence of the contracted power method with purification is illustrated with several molecules. While providing a new tool for semiempirical, Hartree–Fock, and density functional calculations, the 1,2-CSE also represents an initial step toward a linear-scaling algorithm for solving higher CSEs which explicitly treat electron correlation.
A recent advance in the theory of the contracted Schr\"odinger equation (CSE), in which only the anti-Hermitian part of the equation is solved, permits the direct determination of ground-state two-electron reduced density matrices (2-RDM's) that yield 95%--100% of the correlation energy of atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schr\"odinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDM's, the role of Nakatsuji's theorem, and the structure of the wave function. The ACSE is also formulated in the Heisenberg representation and related to canonical diagonalization. The solution of the ACSE is illustrated with a variety of molecules including H2O{\mathrm{H}}_{2}\mathrm{O}H2​O, CH2{\mathrm{CH}}_{2}CH2​, NH4+{{\mathrm{NH}}_{4}}^{+}NH4​+, HF\mathrm{HF}HF, and N2{\mathrm{N}}_{2}N2​, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-$\ensuremath{\zeta}$ basis set. The computed 2-RDM's very closely satisfy known NNN-representability conditions.
The theory of the contracted Schr\"odinger equation (CSE) [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)] is connected with traditional methods of electronic structure including configuration-interaction (CI) and coupled-cluster (CC) theory. We derive a transition contracted Schr\"odinger equation (TCSE) which depends on the wave function $\ensuremath{\psi}$ as well as another N-particle function $\ensuremath{\chi}$ through the two-, three-, and four-particle reduced transition matrices (RTMs). By reconstructing the 3 and 4 RTMs approximately from the 2-RTM, the indeterminacy of the equation may be removed. The choice of the reconstruction and the function $\ensuremath{\chi}$ determines whether one obtains the CI, CC, or CSE theory. Through cumulant theory and Grassmann algebra we derive reconstruction formulas for the 3- and 4-RTMs which generalize both the reduced density matrix (RDM) cumulant expansions as well as the exponential ansatz for the CC wave function. This produces a fresh approach to CC theory through RTMs. Two theoretical differences between the CC and the CSE theories are established for energetically nondegenerate states: (i) while the CSE has a single exact solution when the 3- and 4-RDMs are N-representable, the CC equations with N-representable 3- and 4-RTMs have a family of solutions. Thus, N-representability conditions offer a medium for improving the CSE solution but not the CC solution, and (ii) while the 2-RDM for an electronic Hamiltonian reconstructs to unique N-representable 3- and 4-RDMs, the 2-RTM builds to a family of N-representable 3- and 4-RTMs. Hence, renormalized reconstructions beyond the cumulant expansion may be developed for the 2-RDM but not for the 2-RTM without explicit use of the Hamiltonian. In the applications we implement our recently developed reconstruction formula for the 3-RDM which extends beyond the cumulant approximation. Calculations compare the 3-RDM and 3-RTM reconstructions for the molecules LiH, BeH2,{\mathrm{BeH}}_{2},BeH2​, BH3,{\mathrm{BH}}_{3},BH3​, and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O as well as for systems with more general two-particle interactions. The TCSE offers a unified approach to electronic structure.
Atomic and molecular ground-state energies are variationally determined by constraining the two-particle reduced density matrix (2-RDM) to satisfy positivity conditions. Because each positivity condition corresponds to correcting the ground-state energies for a class of Hamiltonians with two-particle interactions, these conditions collectively provide a new approach to many-body theory that, unlike perturbation theory, can capture significantly correlated phenomena including the multireference effects of potential-energy surfaces. The D, Q, and G conditions for the 2-RDM are extended through generalized lifting operators inspired from the formal solution of N-representability. These lifted conditions agree with the hierarchy of positivity conditions presented by Mazziotti and Erdahl [Phys. Rev. A 63, 042113 (2001)]. The connection between positivity and the formal solution explains how constraining higher RDMs to be positive semidefinite improves the N representability of the 2-RDM and suggests using pieces of higher positivity conditions that computationally scale like the D condition. With the D, Q, and G conditions as well as pieces of higher positivity the electronic energies for Be, LiH, H2O,{\mathrm{H}}_{2}\mathrm{O},H2​O, and BH are computed through a primal-dual interior-point algorithm for positive semidefinite programming. The variational method produces potential-energy surfaces that are highly accurate even far from the equilibrium geometry where single-reference perturbation-based methods often fail to produce realistic energies.
Through the 3,5-contracted Schrödinger equation (3,5-CSchE) quantum energies and 3-particle reduced density matrices (3-RDMs) are determined directly without wave functions. Since the 3,5-CSchE involves the 5-RDM, its solution is indeterminate without N-representability conditions. However, the indeterminacy of the 3,5-CSchE may be removed through a reconstruction strategy for building the 4- and 5-RDMs from the 3-RDM. We present a systematic procedure for obtaining corrections for Valdemoro's reconstruction functionals from two complementary approaches, the particle–hole duality and the theory of cumulants. With the cumulants we are able to demonstrate that we have obtained all terms in the reconstruction functionals which may be written as antisymmetric products of the lower rdms. The cumulants allow us to understand the reconstruction functionals in terms of a renormalized many-body perturbation theory. The reconstruction functionals also lead to a natural generalization of Wick's theorem for evaluating expectation values of fermionic annihilation and creation operators with respect to correlated reference states. Previous work [Phys. Rev. A 57, 4219 (1998)] has explored the determination of correlation energy and 2-RDMs through the 2,4-CSchE, also known as the density equation. Because the reconstruction functionals employed with the 3,5-CSchE depend only on the antisymmetric products of lower RDMs in constrast to those used with the 2,4-CSchE, the 3,5-CSchE method presented here does not require the solution of systems of linear equations during reconstruction or the storage of the reconstructed RDMs. Application of the 3,5-CSchE technique to a quasi-spin model generates ground-state energies and 2-RDMs similar in accuracy to single–double configuration interaction (SDCI). We employ a simple iterative procedure for the solution of the 3,5-CSchE without traditional diagonalization. The CSchE techniques offer an approximate solution of the N-representability problem and a new approach to electron correlation. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 557–570, 1998